Left Regular Representation is Transitive Group Action

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Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $*: G \times G \to G$ be the group action:

$\forall g, h \in G: g * h = \map {\lambda_g} h$

where $\lambda_g$ is the left regular representation of $G$ with respect to $g$.


Then $*$ is a transitive group action.


Proof

Let $g, h \in G$.

Then:

\(\ds \exists a \in G: \, \) \(\ds h\) \(=\) \(\ds g \circ a\) Group has Latin Square Property
\(\ds \leadsto \ \ \) \(\ds h\) \(=\) \(\ds \map {\lambda_g} a\) Definition of Left Regular Representation
\(\ds \leadsto \ \ \) \(\ds h\) \(=\) \(\ds g * a\) Definition of $*$

$h$ is arbitrary, therefore the above holds for all $h \in G$.

By definition of orbit:

$\Orb h = \set {t \in G: \exists g \in G: g * h = t}$

That is:

$\Orb h = G$

Hence the result by definition of transitive group action.

$\blacksquare$


Also see


Sources

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